Beam System Results

Multiple degree of freedom spring / mass systems are excellent tools for understanding the overall behavior of mechanical systems. Unfortunately, real systems typically can not be modelled with just simple spring / mass subsystems. Therefore, to verify that the concepts of equivalent mass and focused modal energy transfer also work with non-spring / mass systems, two cantilever beam models are used: one with a beam clamped to a non-moving base and impacted at the free end of the beam and one with a beam clamped to a moving one dimensional base. A similar beam model from section 2.4.2 was used for the numerical verification. The modelled beam is not a realistic energy generation device for energy harvesting, but it does have low frequency dynamics that helps with computational efficiency.

4.6.1. Non-Moving Base

To demonstrate the use of equivalent mass of the impact point with a real structure, a state space model generated from the results of a one dimensional piezoelectric cantilever beam finite element analysis model (similar to section 2.4.2; see Table 3 for properties) is used. It is assumed that the impact occurs at the tip of the cantilever beam (i.e., the free end). Unlike the previous spring / mass systems, because of the impact location, a non-zero moment arm now exists – the length of the beam; therefore, the equation of equivalent mass (Eq. 52) can not be used. Unfortunately, there is no simple, closed form formula to obtain the equivalent mass for the cantilever beam. The

equivalent mass varies greatly as a function of the length of the beam. A very short beam – a very stiff structure – can be treated by assuming the beam is a rigid body rotating about a fixed point. Then, conservation of linear momentum, angular momentum, and kinetic energy can be used to estimate the equivalent mass of the beam. However, as the length of the beam increases, the flexibility of the beam causes the rigid body estimate to no longer be valid. A simple one dimensional numerical search can be conducted to find the beam’s equivalent mass, and, therefore, the optimal mass of the impactor.

Table 3 – Properties for cantilever shunted piezoelectric beam model Parameter Beam Value _{(top and bottom of beam)} Piezo Value

length 36" 18"

width 3" 3"

thickness 1/8" 1/16"

density 0.29 lb/in3 _{0.29 lb/in}3

modulus 29,000 ksi See Eq. 43

To simulate an electrical load for the piezoelectric device, a simple resistor shunt was connected to the piezoelectric device (Hagood et al., 1990). The resistance of the resistor was tuned to obtain the maximum amount of damping to the cantilever beam (maximum energy output) for the first bending mode. The response of the system during the impact and after the rebound of the impactor are presented in Figure 31 and Figure 32. The kinetic and potential energies of the system and impactor are presented in Figure 33.

Figure 31 – Dynamic response of impactor and beam during impact: (a) impactor (sphere) and impact point (end of beam), and (b) 30%, 60%, and 90% of the beam’s total length

Figure 32 – Dynamic response of impactor and beam after impact: (a) impactor (sphere) and impact point (end of beam), and (b) 30%, 60%, and 90% of the beam’s total length

Just as with the single and multiple degree of freedom system cases, Figure 31(a) shows that at the end of the impact (approximately 44 µs), the impact spring is no longer deformed; therefore, the simulation of the impact stops and releases the impactor from the system. Also note that again, just as in the single and multiple degree of freedom system cases, at the end of the impact, the impactor has zero velocity; all of its initial kinetic energy has been transferred into the system. From Figure 31(b), it is clear that the impact occurs over a timespan that is much too short for the rest of the system to respond – the length of the beam ensures that the effect of the impact is a localized effect. However, this “slowness” to respond is a characteristic of the system properties, especially the beam’s length and stiffness, and will depend on the actual system being impacted. Eventually, as

shown in Figure 32(a) and (b), the entire system responds to the impact at the end of the beam. Figure 33 is again very similar to energy transfer that is seen for the single and multiple degree of freedom system cases – the kinetic energy from the impactor has been transferred to the entire system; the energy being shared between the modes of the system. The final amount of energy obtained by each mode is dependent on the system properties, the location of the impact, and the shape of the impact force applied to the system. Note that approximately 90% of the impact’s energy was transferred into the higher modes of the system (> 200 Hz). Since the electrical load (piezoelectric device resistor shunt) was tuned to dissipate energy at the first bending mode of the system (approximately 5 Hz), very little energy, approximately 10% of the impact energy, was used by the load (or dissipated by the system) over the 30 seconds of simulation time. As shown in

section 4.5.1, focusing the impact energy into a mode that is most efficient for energy production will provide a more efficient system for an energy harvester.

Figure 33 – System energy: (a) during the impact and (b) after the impact – shunt energy denotes energy used by electrical load

4.6.2. One Dimensional Moving Base

To demonstrate the ability to focus impact energy to a mode of interest of a real structure, the same state space model used above (section 4.6.1) was used; however, it was modified such that the cantilever beam was clamped to a one dimensional spring / mass system (see Figure 34), which was tuned to the first bending mode of the cantilever beam – the same mode the resistor shunt was tuned, and acted as the impact point. The mass of the base was set to 100 times the total mass of the piezoelectric beam, ensuring that the impactor, base, and beam system had the behavior of a

simple spring / mass system – thus allowing the use of Eq. 52 to obtain the equivalent mass of the system. The response of the system during the impact and after the rebound of the impactor are presented in Figure 35 and Figure 36. The kinetic and potential energies of the system and impactor are presented in Figure 37.

Figure 34 – Piezoelectric cantilever beam with base system – impact on base

Similar to the results of section 4.5.1, Figures 35, 36, and 37 demonstrate that attaching an energy generation device (e.g., piezoelectric beam) to a base spring / mass system tuned to the optimal energy producing mode of the system produces the desired results. Figure 37(a) clearly shows that nearly all of the impact energy was transferred into the first bending mode of the piezoelectric beam, and that the resistor shunt, Figure 37(b), was able to use (or dissipate) 90% of the generated energy over the 30 seconds of simulation time.

Figure 35 – Dynamic response of impactor, base, and beam during impact: (a) impactor (sphere) and impact point (end of beam), and (b) 30%, 60%, and 90% of the beam’s total length

Figure 36 – Dynamic response of impactor, base, and beam after impact: (a) impactor (sphere) and impact point (end of beam), and (b) 30%, 60%, and 90% of the beam’s total length

Figure 37 – System energy: (a) during the impact and (b) after the impact – shunt energy denotes energy used by electrical load